# Power in physics: its formula, units, examples, types of power and we put solved exercises

We explain what power is, its formula, units, examples, types of power and we put solved exercises. Examples of power in physics

The **power** in physics is a scalar quantity used to indicate the speed with which work is done, or dispensed or consumed energy. It is the key factor to determine how efficient a machinery is, as well as to optimize energy consumption.

For example, a lawn mower can do the job of cutting grass in half an hour or maybe two hours. Clearly, the mower that gets the job done in less time develops more power, because it does the same job faster. Examples of power in physics

In general, power is expressed as:

**Formulas Examples of power in physics**

First, the mean power P _{m} is defined as the quotient between the amount of work done ΔW and the time Δt it took to do it:

In the International System of SI Units, power is measured in **joule / second** , a unit called watt or watt, to honor the Scottish engineer James Watt (1736-1819), who contributed to the development of the steam engine. Examples of power in physics

- Horsepower (CV).
- Horse power (hp or
*horse**power*). - Ergs / second.
- Foot ∙ pound / second.
- Calories / second.
- Kilogram / second.
- Btu / hour.

Some equivalences are given below:

- 1 hp = 550 foot ∙ pound / second = 745.7 W = 2545 Btu / h

The kilowatt-hour that appears frequently in the electricity bill is not a unit of power, but of energy, as well as the Btu or British Thermal Unit, a unit widely used in the field of refrigeration and air conditioning. Examples of power in physics

The instantaneous power P is calculated taking a very small time interval. Making Δt → 0 in the average power it is transformed into the instantaneous power, which is then expressed as the derivative of the work with respect to time: Examples of power in physics

**Efficiency**

The efficiency of machinery is measured by comparing the useful work it does with the energy required to start it.

It happens that no matter how perfect a machinery is, it will never transform all the energy that is contributed to it into useful work. When there are moving parts, friction is responsible for transforming one part into heat and another probably into sound, which are not used. Examples of power in physics

As explained before, the mechanical efficiency ε is as:

With W _{s} the output work and E _{e} the input energy. By multiplying by 100%, the percentage efficiency is obtained, which can also be found through the quotient between the input power P _{e} and the output power P _{s} :

For example, if a machine is 45% efficient, it means that only 45% of the energy input serves the purpose of the machine, and the remaining 55% is lost in heat, sound, or some other form of energy. Examples of power in physics

__Power types__

__Power types__

Power can be developed by different kinds of forces, thus there is mechanical power, associated with moving objects, electrical power, sound power, thermal power and more.

**Mechanical power**

A moving object develops a power related to its speed. Since work is defined as the scalar product of force and displacement, an instantaneous power is produced given by:

Where:

By substituting this equation in the previous one, it is obtained that the mechanical power is the scalar product between the force vector and the velocity vector: Examples of power in physics

**Electric power Examples of power in physics**

It is the rate at which a source or battery delivers energy, at a point load q. This delivery may not be done at a constant rate, so an average power is defined:

Where V is the voltage and Δt is the time interval. If the current and voltage are constant in time, the current is:

I = q / Δt

And the power is also constant, expressing itself as:

P = I ∙ V

**Sound power**

Sound waves carry energy as they propagate, thanks to the pressure on the particles in the medium. For them, the concept of sound intensity is used, which is the power per unit area, which is measured in W / m ^{2} in SI: Examples of power in physics

The total power is calculated by means of the integral over the surface S:

The surface could be, for example, a sphere of radius r.

**Thermal power**

It is the rate at which a certain system releases energy in the form of heat: Examples of power in physics

**Stefan-Boltzmann law**

For the heat that is transmitted by radiation, the Stefan-Boltzmann law is valid:

Where:

- T is the temperature in kelvin.
- σ is the Stefan-Boltzmann constant: σ = 5.67 × 10
^{8}W / (m^{2}K^{4}). - The emissivity of the material is e, whose value is between 0 and 1 and is specific to each material.
- A is the surface area of the body. Examples of power in physics

__Examples of power in physics__

__Examples of power in physics__

**Air conditioners and heaters**

Air conditioning and heating equipment are classified by their power. Manufacturers and designers have empirical formulas to calculate the power that equipment must have to adequately air-condition a room with certain dimensions.

**Light bulbs Examples of power in physics**

People often use the power of an electric bulb to find out how bright it is.

**Home appliances**

All electrical appliances indicate on their labels the power they consume to fulfill their function.

**Machinery Examples of power in physics**

Machinery in general, such as cranes for example, are classified according to the power they develop to lift weights.

**Mechanical ventilators for ICU patients**

The power of a mechanical ventilator for people in intensive care is carefully monitored to avoid lung injury to patients.

__Training__

__Training__

**Exercise 1**

A girl eats the equivalent of 8.4 × 10 ^{6} J daily to keep her weight constant . How much is the average power you develop per day worth?

**Solution Examples of power in physics**

Average power is the work done per unit of time. If the girl keeps her weight constant, she expends all the energy from food to do useful work. A day has 24 hours, which is equivalent in seconds to:

1 day = 24 h = 86400 s

**Exercise 2**

A sports car with a mass of 1500 kg is capable of accelerating from 0 to 90 km / h in 5.0 s. What average power is required to achieve this acceleration? Examples of power in physics

**Solution**

The formula for mechanical power is used:

Force is calculated using Newton’s second law:

The average acceleration of the car, assuming the motion is uniformly varied rectilinear, is:

Before substituting, a unit conversion is carried out:

90 km / h = 25 m / s

Force and displacement are in the same direction, so power is simply the product of the magnitude of the force and the average velocity: Examples of power in physics

Substituting in the power:

P = 1500 kg × 5m / s ^{2} × 12.5 m / s = 93 750 W